Whether Cisco, HP or another provider, with any form of link aggregation, multiple physical ports are treated as a single logical port, after all that is the whole point of link aggregation. Just be careful of terminology used by OSes such as bonding or teaming since they may or may not be doing link aggregation depending on the configuration.
As to running STP, as Ricky and Olipro have said, by all means you should run it. Which flavor depends on your need, however generally keep to a "standards" based protocol. With only two switches (and probably not that many VLANs), I would think you would be fine with RST. On any larger networks, I would recommend MST, but that may be a bit of overkill in this situation and without understanding it properly you may end up creating problems.
While you may not be concerned about more than the two switches, who knows what the future will hold. Also, it helps prevent a number of problems that can be created by users.
For instance, I have run across several instances where users on a network bring in their own dumb switch to connect additional devices in their office/cubicle and decide that if one connection to the wall is good, then two must be better.